Exponential Growth
The growth of a given variable y with an initial value of Po can be modeled with an exponential function as follows:
[tex]y=P_o(1+r)^t[/tex]
Where r is the growth rate and t is the time.
In 2006 (here we assume t = 0), the number of cell phone subscribers was Po = 233 million. We know the number of subscribers increases by r = 6% every year.
a.
With the given information, we can write the exponential function. We must express the rate in decimal, thus r = 6/100 = 0.06.
Our model is expressed as:
[tex]\begin{gathered} y=233(1+0.06)^t \\ Operate\colon \\ y=233(1.06)^t \end{gathered}[/tex]
This is the exponential model.
The number of subscribers in 2008 can be estimated with our model. The value of t is t = 2008 - 2006 = 2 years:
[tex]\begin{gathered} y=233(1.06)^2 \\ y=233\cdot1.1236 \\ y=262 \end{gathered}[/tex]
The estimated number of subscribers in 2008 is 262 million.
b.
It's required to find the value of t when y = 278. Substituting in the model function:
[tex]278=233(1.06)^t[/tex]
We need to solve for t. Dividing by 233:
[tex]\frac{278}{233}=1.06^t[/tex]
Taking natural logarithms on both sides:
[tex]\begin{gathered} \ln \frac{278}{233}=\ln 1.06^t \\ \ln \frac{278}{233}=t\ln 1.06 \\ t=\frac{\ln \frac{278}{233}}{\ln 1.06} \end{gathered}[/tex]
Calculating:
t = 3 years.
In 2009, the number of cell phone subscribers was about 278 million
